Chief Justice Roberts to the Rescue?

Suppose you were the Chief Justice of the Supreme Court. The President has nominated a justice to the Court who you would like to see on the court, and potentially close to 100 members of the U.S. Senate would vote to confirm the nominee under normal circumstances. However, in today's tactical politics, the Senate Majority Leader refused to hold hearings on any nomination, sight unseen. That leaves you in the short term concerned with forging agreement case by case to avoid the unsatisfying 4-4 split decisions. Suppose you regard the President's choice as someone who would add value to the discussion and help forge agreement in difficult cases, so the Senate has left your court hobbled. Not only that, but you even made a recent speech indicating that the nomination process "is not functioning very well" and that "We [on the Court] don't work as Democrats or Republicans", but it is getting harder and harder to maintain that perception among the general public. Is there any feeling of being used by that other branch? Sure, Judge Garland was called a political pawn by some, but isn't the Court being used as a political pawn. Is there anything a Chief Justice can do?
Well...one countertactic would be to rule on any case on which the four liberal justices agree, automatically with those justices to forge a majority of five so long as the Garland nomination languishes without consideration. Even signaling that intent could prompt the Senate to move quickly to consider the nomination. Of course, any of the four conservative justices could take this approach, but the Chief Justice bears the greatest responsibility to protect the institution. Obviously, this is outside of normal judicial temperament and so unlikely, but when one considers that an 8-justice court split 4-4 could persist for years, one wonders if the Chief Justice is at all tempted. Sure, the House could threaten to begin impeachment proceedings, but that is even more unlikely.